We identify and roughly characterize 66 candidate binary star systems in the Pleiades, Praesepe, and NGC 2264 star clusters based on robotic adaptive optics imaging data obtained using Robo-AO at the Palomar 60″ telescope. Only $\sim$10% of our imaged pairs were previously known. We detect companions at red optical wavelengths having physical separations ranging from a few tens to a few thousand AU. A 3-sigma contrast curve generated for each final image provides upper limits to the brightness ratios for any undetected putative companions. The observations are sensitive to companions with maximum contrast $\sim$6$^m$ at larger separations. At smaller separations, the mean (best) raw contrast at 2 arcsec is 3.8$^m$ (6$^m$), at 1 arcsec is 3.0$^m$ (4.5$^m$), and at 0.5 arcsec is 1.9$^m$ (3$^m$). PSF subtraction can recover close to the full contrast in to the closer separations. For detected candidate binary pairs, we report separations, position angles, and relative magnitudes. Theoretical isochrones appropriate to the Pleiades and Praesepe clusters are then used to determine the corresponding binary mass ratios, which range from 0.2-0.9 in $q=m_2/m_1$. For our sample of roughly solar-mass (FGK type) stars in NGC 2264 and sub-solar-mass (K and early M-type) primaries in the Pleiades and Praesepe, the overall binary frequency is measured at $\sim$15.5% $\pm$ 2%. However, this value should be considered a lower limit to the true binary fraction within the specified separation and mass ratio ranges in these clusters, given that complex and uncertain corrections for sensitivity and completeness have not been applied.